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A NEW STOCHASTIC BACKSCATTER
MODEL FOR LARGE-EDDY SIMULATION
OF NEUTRAL ATMOSPHERIC FLOWS
JAMES JOSEPH O’NEILL
A thesis submitted to the University of Birmingham for the degree of DOCTOR
OF PHILOSOPHY
School of Geography, Earth and Environmental Sciences (GEES) College of Life and Environmental Sciences (LES) University of Birmingham June 2016
University of Birmingham Research Archive
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Abstract
A stochastic backscatter (SB) approach to subgrid-scale (SGS) modelling for large-eddy simulation (LES) of the neutral atmospheric boundary layer (ABL) has previously been shown to reduce excessive velocity shear, as seen with the popular Smagorinsky SGS model, in the under-resolved surface layer. However, previous SB models exhibit unwanted grid-dependency issues, and the range of atmospheric flows tested remains limited. Here, a new SB model is proposed that uses a grid-adaptive filter to control the length-scale, anisotropy and momentum flux of the backscatter fluctuations, independently of the model grid. Model performance is confirmed to be grid-independent in simulations of the neutral ABL, in which an 80% reduction in excessive near-surface velocity shear is achieved.
The model is also applied to street canyon flow, where the shear layer that separates the recirculating vortex within the canyon from the external flow is again typically under-resolved in most LES set-ups. The backscatter acts to increase momentum transfer across the shear layer, bringing the simulated vortex intensity significantly closer towards wind-tunnel observations. A passive tracer is also released to model traffic emissions, and the pollutant exchange velocity between the canyon and the external flow is again found in better agreement with wind-tunnel data. This information can be used to improve operational urban dispersion models.
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For Andreea
Big whorls rise from little whorls Through merging of the latter, And big whorls merge to greater whorls
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Acknowledgements
I am enormously grateful to my primary supervisor, Xiaoming Cai, and my secondary supervisor, Rob Kinnersley, for their continued support over these last 3.5 years. Xiaoming, you were always available when I needed help, nurturing towards my academic improvement, and quick and thorough in responding to my (many) queries. Rob, our progress meetings provided the ideal setting for lucid exchanges of thought, and always acted to remind me of the bigger picture of this research.
I am also very grateful to the UK Natural Environment Research Council and the English Environment Agency for their financial support of this research.
I would like to personally thank a few of my fellow research colleagues, namely Tasos Matsikaris for proof-reading much of my work, Mike Grocott for his moral support (mostly through the purchasing of beer), Sally Rangecroft for her thesis tips, and the rest of Room 425 (and adopted others) for countless enjoyable lunch and tea breaks.
The computations described herein were performed using the University of Birmingham’s BlueBEAR HPC service (http://www.bear.bham.ac.uk).
Finally, to my family, friends and girlfriend, who supported me at every step along the way, I thank you all from the bottom of my heart.
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Table of Contents
List of figures ... viii
List of tables ... x
List of symbols ... xi
List of abbreviations ... xv
1. INTRODUCTION ... 1
2. LITERATURE REVIEW ... 6
2.1 Neutral atmospheric flows ... 6
2.1.1 Foundations ... 6
2.1.2 Backscatter... 8
2.1.3 The neutral ABL ... 13
2.1.4 Street canyon flow ... 14
2.2 Large-eddy simulation ... 16
2.2.1 Foundations ... 16
2.2.2 Subgrid-scale modelling ... 20
2.2.3 Previous LES of the neutral ABL ... 23
2.2.4 Previous LES of street canyon flow ... 26
2.3 Backscatter modelling ... 30
2.3.1 Deterministic backscatter models ... 31
2.3.2 Stochastic backscatter models ... 33
2.3.3 SB modelling within other communities ... 39
2.4 Research objectives ... 42
3. THE NEW STOCHASTIC BACKSCATTER MODEL ... 48
3.1 Introduction ... 48
3.2 SB model foundations ... 49
3.3 New grid-adaptive filter ... 52
3.3.1 Filter kernel ... 52 3.3.2 Filter shape ... 53 3.3.3 Filter weights ... 53 3.3.4 Filter width ... 54 3.3.5 Level of anisotropy ... 55 3.3.6 Examples of use ... 56
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3.4 New scaling procedure ... 59
3.4.1 Horizontally homogeneous flow ... 59
3.4.2 Horizontally inhomogeneous flow ... 63
3.5 Backscatter length-scale components ... 64
3.6 Backscatter vertical momentum flux ... 66
3.7 Summary and conclusions ... 71
4. THE NEUTRAL ATMOSPHERIC BOUNDARY LAYER ... 74
4.1 Introduction ... 74
4.2 Methodology ... 75
4.2.1 The RAMS LES model ... 75
4.2.2 LES model configuration ... 76
4.2.3 SB model configuration ... 77
4.3 Results and discussion ... 80
4.3.1 Backscatter acceleration fields ... 80
4.3.2 LES fields ... 86
4.3.3 Additional CPU time ... 96
4.4 Summary and conclusions ... 98
5. STREET CANYON FLOW ... 100
5.1 Introduction ... 100
5.2 Methodology ... 101
5.2.1 LES model configuration ... 101
5.2.2 SB model configuration ... 105
5.2.3 Wind-tunnel data ... 109
5.3 Results and discussion ... 110
5.3.1 Primary eddy intensity ... 110
5.3.2 Turbulent kinetic energy ... 117
5.3.3 Shear layer entrainment: air exchange rate... 119
5.4 Summary and conclusions ... 121
6. STREET CANYON DISPERSION ... 125
6.1 Introduction ... 125
6.2 Methodology ... 126
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6.2.2 SB model configuration ... 128
6.3 Results and discussion ... 129
6.3.1 Model validation: Pollutant exchange velocity ... 129
6.3.2 Model validation: Wall concentration profiles ... 132
6.3.3 Mean 2-D fields: Pollutant concentration ... 134
6.3.4 Mean 2-D fields: Turbulent pollutant flux ... 137
6.3.5 Pollutant exchange rate ... 140
6.4 Summary and conclusions ... 143
7. CONCLUSIONS ... 145
7.1 Summary of main findings ... 145
7.2 Limitations and further work ... 149
APPENDIX: FORTRAN CODE EXTRACTS ... 153
Publications arising from this thesis ... 157
viii
List of figures
Figure 2.1 – Demonstration of backscatter from DNS ... 11
Figure 2.2 – Schematic of skimming flow ... 15
Figure 2.3 – Schematic of TKE spectrum ... 19
Figure 2.4 – Examples of the ‘overshoot’ ... 25
Figure 2.5 – LES spin-up time from bulk TKE ... 38
Figure 2.6 – Schematic of new SB model vs. previous models ... 44
Figure 3.1 – Grid-adaptive filter weights ... 57
Figure 3.2 – Filtered fields on fixed-aspect-ratio grid ... 58
Figure 3.3 – Filtered fields on stretched vertical grid... 59
Figure 3.4 – Backscatter stresses ... 71
Figure 4.1 – Backscatter anisotropy and length-scale profiles ... 80
Figure 4.2 – Backscatter acceleration fields ... 82
Figure 4.3 – Backscatter acceleration variance profiles ... 84
Figure 4.4 – Backscatter acceleration divergence profiles ... 86
Figure 4.5 – Velocity shear profiles ... 87
Figure 4.6 – Profiles from WM08 paper ... 92
Figure 4.7 – Velocity shear time-series ... 93
Figure 4.8 – Velocity variance profiles ... 95
Figure 4.9 – Bulk TKE time-series ... 97
Figure 5.1 – Schematic of LES domain ... 103
Figure 5.2 – Unfiltered and filtered dissipation field ... 108
Figure 5.3 – Wind-tunnel photo ... 110
Figure 5.4 – Vertical velocity profiles ... 111
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Figure 5.6 – Primary eddy intensities ... 115
Figure 5.7 – RS-TKE profiles... 119
Figure 6.1 – LES domain (with sources) ... 127
Figure 6.2 – Wash-out curves ... 130
Figure 6.3 – Wall concentration profiles ... 133
Figure 6.4 – Concentration fields ... 136
Figure 6.5 – Turbulent pollutant flux fields ... 138
Figure 6.6 – Turbulent momentum flux fields ... 140
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List of tables
Table 4.1 – LES grids ... 77
Table 4.2 – SGS models ... 78
Table 5.1 – SB model configurations ... 107
Table 5.2 – Primary eddy intensities ... 113
Table 5.3 – ACH values ... 121
Table 6.1 – SB model configuration ... 128
Table 6.2 – Exchange velocities ... 132
Table 6.3 – Mean concentrations ... 137
xi
List of symbols
Symbol Description [units]
𝒂, 𝑎𝑖 Backscatter acceleration field (vector, components) [m s-2] 𝐴 Filter weights coefficient
𝐴SC Area of street canyon opening (in plan-view) [m2]
𝛼 Constant (0 ≤ 𝛼 ≤ 1) to control correlation between 𝜙̂𝑥 and 𝜙̂𝑧 𝛼sgs SGS eddy-diffusivity [m2 s-1]
𝐵r Theoretical energy backscatter rate [m2 s-3] 𝐵̂r Actual (modelled) energy backscatter rate [m2 s-3] 𝐶S Smagorinsky constant
𝐶ϵ Constant in dissipation equation 𝐶B Backscatter coefficient (momentum) 𝐶Bθ Backscatter coefficient (scalars) 𝐶 Scalar concentration [μg m−3]
𝐶0 Quasi-steady scalar concentration [μg m−3]
𝐶can Mean concentration within the street canyon [μg m−3] Δ Local grid scale [m]
Δ𝑥, Δ𝑦, Δ𝑧 Grid spacing in 𝑥, 𝑦, 𝑧 [m]
Δeq Equivalent grid scale (Δeq = (Δ𝑥Δ𝑦Δ𝑧)1/3) [m] ΔAR Grid aspect ratio (∆𝑥/∆𝑧)
∆−, ∆+ Distance to adjacent grid point in –/+ direction [m] Δ𝑡 Model time-step [s]
ΔSL Surface layer depth [m]
𝐸 Turbulent kinetic energy (per unit mass) [m2 s-2] 𝜖 Dissipation rate [m2 s-3]
𝑓 Coriolis parameter [s-1]
𝝓̂ , 𝜙̂𝑖 Filtered random field (vector, components) [m
2 s-2]
𝝓, 𝜙𝑖 Scaled, filtered random field (vector, components) [m2 s-2] ΦM Mean nondimensional velocity shear
xii 𝑔𝑖,𝑗,𝑘 Point-wise scaling factor at grid point 𝑖, 𝑗, 𝑘
ℎ(𝑥, 𝑦, 𝑧) 3-D filter kernel
ℎ𝑥(𝑥) 1-D filter kernel in 𝑥
𝐻 Vertical scaling
height
Neutral ABL flow: Building height [m]
Street canyon flow: Boundary layer scaling height [m]
𝑖, 𝑗, 𝑘 Indices of discrete model grid-points in 𝑥, 𝑦, 𝑧
𝑘 Wave-number
𝜅 Von-Kármán constant
𝑙 SGS mixing length [m]
𝑙0 Maximum of 𝑙 in well-resolved regions [m] 𝑙f LES filter width [m]
𝑙B Backscatter length-scale [m]
𝑙B𝜁 Backscatter length-scale component (filter width) in 𝜁 dimension [m] 𝑙̂B Backscatter length-scale in the flow interior [m]
𝐿𝑥, 𝐿𝑦, 𝐿𝑧 Domain extent in 𝑥, 𝑦, 𝑧 [m]
𝜆 Tuning parameter of 𝒪(1) to control backscatter length-scale 𝑀sc Mean pollutant mass within the street-canyon [μg]
𝑀bg Mean background pollutant mass (outside the street-canyon) [μg] 𝑛 Exponent in SGS mixing length equation
𝑁𝑥, 𝑁𝑦, 𝑁𝑧 Number of grid points in modelling domain in 𝑥, 𝑦, 𝑧
𝜈 Kinematic viscosity [m2 s-1] 𝜈𝐶 Molecular diffusivity [m2 s-1] 𝜔 Vorticity [s-1]
𝜔PE Primary eddy intensity metric [s-1]
𝑝 Pressure [Pa]
𝑄s Single line-source emission rate [μg m−1 s−1] 𝑄 Total line-source emission rate [μg m−1 s−1] 𝑄tot Total emission rate [μg s−1]
𝒓, 𝑟𝑖 Random number field (vector, components) [m2 s-2]
Re Reynolds number
xiii 𝜌𝑖,𝑗,𝑘∆𝑥 Autocorrelation coefficient between 𝜙̂𝑖,𝑗,𝑘(𝑥, 𝑦, 𝑧) and 𝜙̂𝑖,𝑗,𝑘(𝑥 + Δ𝑥, 𝑦, 𝑧) 𝑆 Shorthand for (ΦM)max
𝑆𝑖𝑗 Rate-of-strain tensor [s-1] 𝑆Δ𝑧 Vertical grid stretch factor
𝑆𝑐 Schmidt number
𝑆𝑟𝑐 Scalar source term [μg m−3 s−1]
𝜎𝑖 Turbulent SGS scalar fluxes [μg m−2 s−1] 𝜎𝑋, 𝜎𝑋2 Standard deviation, variance of property X
𝑡 Time [s]
𝑇 Reference time-scale [s] 𝑇B Backscatter timescale [s] 𝜏 ‘e-folding’ time-scale [s]
𝜏𝑡𝑟 Transition period time-scale in SC flow [s] 𝜏𝑖𝑗 Turbulent SGS stress tensor [m2 s-2]
𝑢, 𝑣, 𝑤 Velocity components in 𝑥, 𝑦, 𝑧 [m s-1]
𝑢∗ Friction velocity [m s-1]
𝑢𝑖′B Backscatter velocity fluctuation in 𝑥𝑖 dimension [m s-1] 𝑢𝑖′𝑢
𝑗′𝐵 Backscatter stresses [m2 s-2] 𝑈ref Reference velocity scale [m s-1] 𝑈g Geostrophic wind speed [m s-1]
𝑈∞ Far-field free-stream velocity in SC flow [m s-1] 𝑣sgs SGS eddy-viscosity [m2 s-1]
𝑣e Pollutant exchange velocity [m s-1]
𝑉 Volume of street canyon (below roof-level) [m3] VMFB Backscatter vertical momentum flux factor
𝑤𝜉 Filter weight at distance 𝜉 normalised by its filter coefficient 𝑊𝜉 Filter weight at grid point at distance of 𝜉 from filter centre
𝑊 Street width [m]
𝑥, 𝑦, 𝑧 Streamwise, spanwise and vertical directions [m]
xiv 〈𝑋〉 Angled brackets denote a horizontal average of property X
𝑋′ Prime denotes fluctuation about mean of property X 𝜉 Distance from filter centre [m]
𝜉−, 𝜉+ Lower and upper bound of Gaussian function integral [m] 𝑧0 Surface roughness [m]
𝑧1 Height of lowest model grid level [m]
𝑧Bmax Maximum height of backscatter acceleration field [m] 𝜁f Location of filter centre in 𝜁 dimension [m]
xv
List of abbreviations
Abbreviation Description
§ Section
1-D / 2-D / 3-D One/Two/Three-dimensional
ABL Atmospheric boundary layer
ACH Air exchange rate [m3s−1]
BC Boundary condition
CA Cellular automaton
CFD Computational fluid dynamics
CFL Courant–Friedrichs–Lewy
CPU Central processing unit
DNS Direct numerical simulation
DRM Dynamic reconstruction model
ECMWF European Centre for Medium-Range Weather Forecasts EDQNM Eddy-damped quasi-normal Markovian
GCM General circulation (or global climate) model
LAM Limited area model
LASD Lagrangian-averaged scale-dependent
LES Large-eddy simulation
MT92 Mason and Thomson (1992)
N/A Not applicable
NBA Non-linear backscatter anisotropy NWP Numerical weather prediction PCH Pollutant exchange rate [μg s−1]
PE Primary eddy
PGF Pressure gradient force
RAMS Regional Atmospheric Modelling System RANS Reynolds-averaged Navier–Stokes
RL Roof level
RMS Root-mean-square
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SB Stochastic backscatter
SC Street canyon
SGS Subgrid-scale
SKEB Stochastic kinetic energy backscatter
SMAG Smagorinsky
TKE Turbulent kinetic energy
VMF Vertical momentum flux
WRF Weather Research & Forecasting
WM08 Weinbrecht and Mason (2008)
WT Wind-tunnel
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1. INTRODUCTION
A deeper knowledge of atmospheric motion offers a number of significant benefits. Among others, it guides our ability to predict future weather events, aids our placement of wind turbines for energy production, and improves our understanding of pollutant transport and dispersion after emission. The physical processes that govern the evolution of atmospheric motion are already well understood, and the set of equations that underpin these processes are thought to be comprehensive and accurate. Unfortunately, however, this set of equations consists of nonlinear partial differential equations (including the Navier–Stokes equations) that have no known analytical solution. As a result, atmospheric modellers are forced to use numerical methods to approximate solutions on discretised model grids. The nonlinear nature of the atmosphere also means that kinetic energy is continuously transferred across the various scales of motion, in a process known as the energy cascade. Thus, in order to faithfully reproduce the exact evolution of any atmospheric state, all scales of motion – from the largest planetary scale to the smallest molecular scale – must be simulated simultaneously. Clearly, the grid resolution required for such a simulation is unfeasible for any modern-day, or future, computer.
As a consequence, atmospheric modellers must decide a priori the most important range of scales (which will be situation-dependent) that are to be explicitly resolved by their model; the upper bound of this range is set by the extent of the modelling domain, and the lower bound by the model’s grid resolution. The influence of any scales greater than the largest resolved scales must then enter the simulation by way of appropriate boundary conditions, and the influence of all scales smaller than the smallest resolved scales – the so-called subgrid scales (SGS) – must be parameterised by the SGS model. The work undertaken in this thesis
2 looks to improve an SGS model used for simulating neutral atmospheric flows from the
boundary-layer scale down to the urban micro-scale. This range of scales is particularly important to us as humans. The atmospheric boundary layer (ABL), defined by Stull (1988) to be the lowest part of the atmosphere that is directly influenced by the presence of the earth’s surface, is the region in which we all live and breathe. The urban micro-scales encompass the flow patterns found within the urban canopy layer, where the street canyon forms the main building block (Oke, 1987). The World Health Organisation (WHO, 2015) estimates that over half the world’s population currently lives in urban areas (a fraction that is set to continue to rise), and so a firm knowledge of how the urban environment affects atmospheric flow and pollutant dispersion is also important.
The boundary-layer and urban micro-scales both fall within the broader turbulence scales, in which individual atmospheric motions, or eddies, are fully three-dimensional (3-D) (as opposed to, for example, the synoptic scale, in which individual structures such as cyclones and anticyclones are quasi 2-D). From the boundary-layer scale to the urban micro-scale, pollutant dispersion will thus be large in both the horizontal and the vertical directions. There are a number of available flow-field and dispersion models for simulating pollutant dispersion at these scales, which cover a range of complexities. Operational models, which must be cost- and time-effective, typically opt for a lower-complexity approach, in which ‘bulk’ parameters determine the level of mixing by atmospheric processes. For example, the Gaussian plume approach, often used for boundary-layer-scale dispersion modelling (e.g. ADMS, Carruthers
et al. (1994)), adopts a bulk parameter to describe plume spread, and the box-model approach, often used for street canyon dispersion modelling (e.g. OSPM, Berkowicz (2000)), adopts a bulk parameter to describe the pollutant mass exchange between the air within the canyon and the boundary-layer atmosphere above it. At the higher end of model complexity spectrum lies
3 the computational fluid dynamic (CFD) modelling approach, in which a numerical solution to the (smoothed) momentum equations is sought directly.
Large-eddy simulation (LES) is a subset of CFD modelling in which all the turbulent scales of motion that can be explicitly resolved by the model grid are prognosed (in space and time). Another, less computationally expensive, approach known as Reynolds-averaged Navier– Stokes (RANS) parameterises all turbulence scales in order to obtain the mean flow-field solution. Although LES requires substantial computational resources that are typically unavailable to operational modellers, it remains a very useful tool in terms of: (i) increasing our understanding of turbulent processes by offering far greater spatiotemporal coverage than that possible by field or laboratory experiment, and (ii) guiding our choice of input parameters for the simpler operational models through calculated bulk statistics from its numerical output. The importance of improving the accuracy of LES is therefore made apparent.
Simulation accuracy is particularly dependent on the SGS model, which must faithfully parameterise the effect of all interactions between the resolved scales of motion and the subgrid (unresolved) scales. The net effect of these interactions is to drain energy from the resolved scales to the unresolved scales, and many SGS models, including the Smagorinsky (1963) model (the most popular SGS model in use in LES today), only seek to represent this net energy drain. However, this net drain is in fact the result of many forward-scatter events (energy transfer from resolved to unresolved scales) and backscatter events (energy transfer from unresolved to resolved scales), which typically occur with similar frequency in any given turbulent flow-field (Geurts, 2004). In many situations, the simplified ‘net drain’ approach adopted by, e.g., the Smagorinsky model can still produce adequate simulation accuracy; however, in other cases – in particular, the cases where backscatter at the grid scale is large – simulation accuracy can be significantly compromised.
4 As a result, more sophisticated SGS models that attempt to represent the forward and backscatter processes separately have also been proposed. One notable approach, pioneered by Mason and Thomson (1992), uses additional stochastic motions to ‘inject’ energy back into the flow at the smallest resolved scales, as a way of mimicking these random backscatter events. This approach has already proven very successful in alleviating discrepancies seen in simulation output with the Smagorinsky model for the case of boundary layer flow, due to its ability to represent important grid-scale backscatter occurring in the under-resolved surface layer. However, two research gaps can be identified, namely that: (i) the existing methodologies for the way in which the backscatter energy is stochastically injected into the modelled flow-field suffer from unwanted grid-dependency issues, meaning that there is scope for the implementation of a new and improved methodology; and (ii) applications of the stochastic backscatter (SB) modelling approach have so far been limited to simulations of boundary-layer flow, and not to other scales such as the urban micro-scale, or, to the author’s knowledge, to simulations that test the subsequent impact on the dispersion of a passive scalar. Both these research gaps are addressed in this thesis, in which it is shown that a new SB methodology can help improve the dynamics and dispersion characteristics in a number of LES applications, including (finally) a fundamental street canyon flow configuration. It is hoped that this will constitute a first step towards providing a more comprehensive database of improved input parameters for operational street canyon models.
The thesis is organised as follows. This introduction (Chapter 1) provides a general overview of the research topic, and is followed by a more in-depth scientific review of the relevant literature to date, ending with a list of clear research objectives (Chapter 2). The following chapter details the methodology behind the new SB model (Chapter 3). The three chapters that follow then present the results from three different applications of the new SB model. The
5 first application is to the previously tested case of the neutral ABL (Chapter 4). This is done in order to compare the new SB model’s performance with that of previous SB models to assess for improvement; in particular, to test the new model’s ability to overcome the grid-dependency issues of the previous models. The second application of the SB model is to the simulation of street canyon flow (Chapter 5), in which model output is compared against that obtained with the Smagorinsky model, using a suitable wind-tunnel dataset for validation. This is followed by a third application of the model, which assesses the subsequent impact on the prediction of traffic-related pollutant dispersion and removal from the street canyon (Chapter 6). Finally, a conclusions section (Chapter 7) summarises the main findings, implications and limitations of the research, as well as suggesting steps for future work.
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2. LITERATURE REVIEW
This literature review is structured as follows. Current knowledge of neutral atmospheric flows (including the physical process of backscatter) is first summarised, with particular attention paid to neutral ABL and street canyon flow (§2.1). The fundamentals of LES and SGS modelling are then covered, followed by a summary of previous LES of the neutral ABL and street canyon flow/dispersion, including issues of simulation accuracy related to the SGS model (§2.2). A more detailed look is then taken at the stochastic backscatter SGS modelling approach, including its advantages and disadvantages over other approaches, similar approaches used within other modelling communities, and current issues and research gaps associated with the technique (§2.3). Finally, the specific objectives of the current research project are identified (§2.4).
2.1 Neutral atmospheric flows 2.1.1 Foundations
Before tackling the literature, a brief theoretical overview of neutral atmospheric flows is first provided. In this thesis, the phrase ‘neutral atmospheric flow’ is used to mean a high-Reynolds-number (Re) (i.e. fully turbulent) flow in which there is no heat energy input, and thus the potential temperature is constant everywhere. In this case, the generation of turbulence occurs through velocity shear alone, which results from the interaction of the flow with a surface (or any other source of friction). Although perfectly neutral atmospheric conditions are rarely observed in reality, so-called ‘near-neutral’ conditions occur frequently and the neutral case is thus still regarded as a useful subject of study. It also provides a simplified starting point from which added complexity can be later explored.
7 If one also assumes a dry and incompressible gas (the latter is typically valid in the lower atmosphere), then the governing equations of motion are completely described by the continuity (conservation of mass) and Navier–Stokes (conservation of momentum) equations. Using tensor notation, these are given (in an inertial frame) by:
𝜕𝑢𝑖 𝜕𝑥𝑖 = 0, (1) 𝜕𝑢𝑖 𝜕𝑡 + 𝑢𝑗 𝜕𝑢𝑖 𝜕𝑥𝑗 = − 1 𝜌 𝜕𝑝 𝜕𝑥𝑖+ 𝜈 𝜕2𝑢 𝑖 𝜕𝑥𝑗𝜕𝑥𝑗, (2)
where 𝑢𝑖 (= 𝑢, 𝑣, 𝑤 for 𝑖 = 1, 2, 3) is the velocity component in the direction 𝑥𝑖 (= 𝑥, 𝑦, 𝑧 for
𝑖 = 1, 2, 3), 𝑡 is time, 𝑝 is pressure, 𝜌 is the air density, and 𝜈 is the kinematic viscosity. The
second term in Eq. (2), the so-called advection term, is the nonlinear term responsible for momentum transfer across scales, i.e. the energy cascade, which occurs both from larger to smaller scales (forward scatter) and from smaller to larger scales (backscatter – see next section, §2.1.2, for more details).
It is also useful to give a few more equations here, which will be quoted at later points in the thesis. Firstly, the equation for the conservation of a passive scalar released into the flow-field: 𝜕𝐶 𝜕𝑡 + 𝑢𝑗 𝜕𝐶 𝜕𝑥𝑗 = 𝜈𝐶 𝜕2𝐶 𝜕𝑥𝑗𝜕𝑥𝑗+ 𝑆𝑟𝑐, (3)
where 𝐶 is the scalar concentration, 𝜈𝐶 the molecular kinematic diffusivity, and 𝑆𝑟𝑐 a scalar source term. Secondly, it is possible to use a Reynolds decomposition and the rules of Reynolds averaging to obtain the governing equations for the mean fluid motion (utilising the existence of a spectral gap (Stull, 1988)). The dependent variables are decomposed as 𝑢𝑖 = 𝑢𝑖 + 𝑢𝑖′, where the overbar denotes an ensemble average (which, if the flow is
8 stationary, is equivalent to a time average) and the prime denotes the turbulent fluctuation from this value. Substituting into Eqs. (1) and (2) leads to:
𝜕𝑢𝑖 𝜕𝑥𝑖 = 0, (4) 𝜕𝑢𝑖 𝜕𝑡 + 𝑢𝑗 𝜕𝑢𝑖 𝜕𝑥𝑗 = −1 𝜌 𝜕𝑝 𝜕𝑥𝑖 −𝜕(𝑢𝑖′𝑢𝑗′) 𝜕𝑥𝑗 . (5)
The final term in Eq. (5) describes the influence of the so-called Reynolds stresses, i.e. momentum fluxes caused by turbulent motions, on the mean flow. For high-Re flows such as in the atmosphere, this is several orders of magnitude larger than the influence of viscous stresses, hence why the term involving 𝜈 in Eq. (2) has now been dropped. Finally, a similar Reynolds decomposition for scalars yields:
𝜕𝐶 𝜕𝑡 + 𝑢𝑗 𝜕𝐶 𝜕𝑥𝑗 = 𝑆𝑟𝑐 − 𝜕 (𝑢𝑗′𝐶′) 𝜕𝑥𝑗 , (6)
where 𝑢𝑗′𝐶′ describes the turbulent scalar flux in the 𝑥𝑗 direction.
2.1.2 Backscatter
Backscatter refers to the physical process in which turbulent energy is transferred from smaller to larger scales. This is in contrast to forward-scatter, which refers to the transfer of energy from larger to smaller scales. Backscatter is perhaps conceptually harder to visualise than forward-scatter – it is easier to conceptualise smaller eddies being generated from the shearing of larger eddies, as eloquently described in the poem by Lewis F. Richardson (1920):
Big whorls have little whorls That feed on their velocity, And little whorls have lesser whorls
9 Although the net effect of turbulence is to drain energy from the larger to the smaller scales, in reality, the frequency of forward and backscatter events occurring within a turbulent flow are typically almost equal (Geurts, 2004). Conceptually, an individual backscatter event may be thought of as two or more small-scale eddies merging to produce larger scales. A riposte to Richardson’s poem might therefore read:
Big whorls rise from little whorls Through merging of the latter, And big whorls merge to greater whorls
And so on with backscatter.
Although the full 3-D picture is far more complex, backscatter may be demonstrated mathematically using a 1-D spectral representation of turbulence. Consider two simple cosine waves 𝑈1 and 𝑈2, with respective wave-numbers 𝑘1 and 𝑘2, i.e.:
𝑈1 = cos 𝑘1𝑥, (7)
𝑈2 = cos 𝑘2𝑥. (8)
Following Pielke (2002), a nonlinear interaction between these two waves can be represented by their product, i.e.:
𝑈1𝑈2 = cos 𝑘1𝑥 cos 𝑘2𝑥 =
1
2[cos(𝑘1+ 𝑘2)𝑥 + cos(𝑘1− 𝑘2) 𝑥], (9)
i.e. two further waves are produced, one with a higher wave-number than the original two waves (𝑘1+ 𝑘2) and, if 𝑘1 < 2𝑘2, one with a lower wave-number than the original two waves (𝑘1− 𝑘2). In the latter case, energy has been transferred from smaller to larger scales, i.e. backscattered. Again, the above analysis only serves as a simplified example of upscale
10 energy transfer; in reality, backscatter within a 3-D turbulent flow-field is a result of a triple interaction between two smaller-scale eddies and a larger-scale one (Schumann, 1995). Chasnov (1991) provided a clear demonstration of backscatter in his direct numerical simulation (DNS) of freely decaying turbulence. The relevant figure from this paper, reproduced below (Figure 2.1), shows how the turbulent kinetic energy (TKE) spectrum changes with time. As there is no turbulence generation, the total TKE in the system must be decreasing with time due to viscous dissipation, i.e. the area under the TKE curve, 𝐸(𝑘, 𝑡), when plotted on linear axes (rather than logarithmic axes, as here) must decrease with time. If only forward-scatter was occurring in the flow, one would expect to see the energy at the largest scales decrease with time as the forward cascade transferred energy across to the smaller scales. However, it is seen that the redistribution of energy occurs in both the forward direction, resulting in the well-known 𝑘−5/3 spectrum at the smaller scales (Kolmogorov, 1941), as well as the reverse direction (backscatter), giving a tendency towards a 𝑘4 spectrum at the larger scales. The above experiment provides an empirical demonstration of backscatter. However, it is also possible to demonstrate analytically that backscatter occurs across a given length-scale via the theoretical equation for the resolved-scale energy budget, which contains one distinct term describing the drain of energy to smaller scales and another distinct term describing the reverse energy transfer (i.e. backscatter) process (Leslie and Quarini, 1979).
11
Figure 2.1 – Demonstration of backscatter from DNS
Taken from Chasnov (1991), Fig. 2: “Time evolution of the energy spectrum computed from the DNS of freely decaying turbulence. Time steps 0-200 by 10 [later curves tend towards 𝒌𝟒 and 𝒌−𝟓/𝟑 spectrum at low and high 𝒌 respectively]; [cut-off wavenumber]
𝒌𝒄 = 𝟏𝟔.”
Consider the ‘cut-off’ wave-number 𝑘𝑐 shown in Figure 2.1 with the vertical line. Schumann (1995) notes that the backscatter contribution from scales of motion smaller than 𝑘𝑐 (i.e. larger wave-numbers) falls off with 𝑘−6; this means that only those scales of motion that are slightly smaller than 𝑘𝑐 contribute to the backscatter across that wave-number. Work has been done to try to quantify the magnitude of backscatter across smaller-scale wave-numbers (i.e. within the ‘inertial subrange’ - see §2.2.1 for more details). Leslie and Quarini (1979) showed that, in isotropic turbulence, the backscatter rate across wave-numbers within this range is proportional to the local dissipation rate, 𝜖, (i.e., the net energy flux towards smaller scales), with the constant of proportionality being of order unity but varying with the type of filter (e.g. cut-off, top-hat, Gaussian) used to separate the smaller and larger scales for a given wave-number. For a cut-off filter, they found that the rate of backscatter was given by
12 turbulence using eddy-damped quasi-normal Markovian (EDQNM) theory, and found a similar relationship with the dissipation rate of 𝐵r = 1.4𝜖 (for an infinite inertial subrange). Soon after, Mason and Thomson (1992) attempted to quantify 𝐵r for the case of wall-bounded shear flow. Using fairly simplified physical and dimensional reasoning to obtain an estimate for the fluctuating stress gradients, they were able to derive the following relationship:
𝐵r = 𝐶B(𝑙 𝑙0)
5
𝜖, (10)
where 𝐶B is the so-called backscatter coefficient (of order unity), and 𝑙 is the mixing length of the sub-filter scales, with a maximum value of 𝑙0. Thus, far from any surface (where 𝑙 = 𝑙0), the direct proportionality between 𝐵r and 𝜖, as derived for isotropic flow, is recovered. However, closer to surfaces (where 𝑙 < 𝑙0), the backscatter rate is strongly attenuated (due to the effect of the 5th exponent). This is consistent, at least in principle, with the previous observation that only scales slightly smaller than the filter scale contribute to backscatter; thus, if these scales are simply not present (due to being suppressed) close to surfaces, then the backscatter there will be negligible.
It is also possible to derive estimates of the backscatter rate across a given length-scale numerically, by performing DNS (i.e. resolving down to the smallest scales of turbulent motion) and then explicitly filtering the DNS data (i.e. averaging the forward and reverse energy contributions over wave-number shells of a specified radius), as done by, e.g., Kosović
et al. (2002). However, DNS-calculated backscatter estimates often deviate from those predicted by theory (Domaradzki and Saiki, 1997). Furthermore, DNS is restricted to relatively low-Re flows because of the requirement to resolve down to the dissipative scales, and so backscatter-rate estimates for atmospheric (high-Re) flows are not practically possible.
13
2.1.3 The neutral ABL
The neutral ABL is an example of a wall-bounded turbulent shear flow. Hinze (1972) and Davidson (2004) both provide general theoretical summaries for this type of flow, and the theory relating more specifically to the ABL can be found in, e.g. Stull (1988) or Garratt (1994). Von Kármán (1931) helped show that within the surface layer (the lowest 100m or so) of the neutral ABL, the mean wind profile (over flat terrain) takes the following logarithmic form:
𝑈(𝑧) =𝑢∗
𝜅 ln (
𝑧 + 𝑧0
𝑧0 ), (11)
where 𝑢∗ is the friction velocity, 𝑧0 the surface roughness, and 𝜅 the von-Kármán constant. The exact value of 𝜅 is debated, and furthermore varies for different types of flow. Businger
et al. (1971) analysed surface layer observations from the famous measurement campaign in Kansas (1968), suggesting a value of 𝜅 = 0.35 in neutral conditions. Frenzen and Vogel (1995) and Andreas et al. (2006) suggest slightly larger values, with an upper limit of
𝜅 = 0.41 proposed by the former. Alternatively, Cai and Steyn (1996) used LES to determine
a value of 𝜅 = 0.36.
The logarithmic wind profile provides a similarity condition against which atmospheric modellers often validate their numerical output. This log-profile is obtained by integrating the velocity shear profile, which itself provides an alternative similarity condition:
𝑑𝑈
𝑑𝑧 =
𝑢∗
𝜅(𝑧 + 𝑧0). (12)
This is typically rearranged into non-dimensional form to give the normalised velocity shear profile, ΦM:
14
ΦM =𝑑𝑈
𝑑𝑧
𝜅(𝑧 + 𝑧0)
𝑢∗ = 1. (13)
Modellers may thus check how closely their profile for ΦM matches the unit profile (Lu and Porté-Agel, 2014); this is a validation tool that will be used for the new SB model later in the thesis (Chapter 4).
2.1.4 Street canyon flow
With over half of the world’s population living in urban areas (WHO, 2015), it is important to understand the effects of the densely built environment on wind flow and pollutant dispersion. Street canyons – the gap formed between two rows of buildings either side of a street – form a key constituent part of the urban fabric (Oke, 1988). A street canyon is characterised by the ratio of the building height, 𝐻, to the street width, 𝑊. Particular concern surrounds the case of vehicular emissions released within deep street canyons (𝐻/𝑊 ≳ 1), which has been observed to lead to poor ventilation, and thus poor air quality, in the real world (DePaul and Sheih, 1985, Xie et al., 2003). An extreme case occurs when the oncoming wind is perpendicular to the street axis; a skimming flow regime is established (Oke, 1987), in which the bulk flow passes over the street canyon, leaving pollutants largely trapped within the canyon and thus susceptible to build up to potentially harmful levels (see Figure 2.2(c)). Figure 2.2(a) and (b) also characterise the ‘isolated roughness flow’ regime (no interaction between adjacent wakes) and ‘wake interference flow’ regime (some interaction but still distinct), respectively, which can occur in shallower street canyons.
15
Figure 2.2 – Schematic of skimming flow
Taken from Oke (1988), Fig. 1: “The flow regimes associated with air flow over building arrays of increasing 𝑯/𝑾.”
In skimming flow, a sharp velocity gradient exists between the relatively fast moving air above the street canyon and the relatively slow moving air inside it. This creates a narrow but energetic shear layer at roof-level, characterised experimentally by a narrow peak in TKE and momentum flux (𝑢′𝑤′) measurements at that height, e.g. Louka et al. (2000), Blackman et al. (2015). This shear layer represents a plane mixing layer, as it is formed at the boundary of two co-directional flows of differing speeds (Letzel et al., 2008). Louka et al. (2000) analysed the TKE budget equation for neutral flow to reveal that the peak in 𝑢′𝑤′ is a result of a maximum in the shear-production term at roof-level.
The slow-moving air within near-unit-aspect-ratio street canyons typically forms a large-scale rotating vortex that continually recirculates the air (and any pollutants) around the canyon (Lee and Park, 1994, Baik and Kim, 1999, Huang et al., 2000). In addition to this primary eddy (PE), smaller secondary counter-rotating vortices typically form near the corners of the canyon (Liu and Barth, 2002, Cui et al., 2004). Advective transport by each isolated eddy acts to smooth out local gradients of scalar concentration within them, whilst scalar transport
16 between each eddy is largely controlled by the slower turbulent diffusion process (Liu and Barth, 2002). For deeper canyons, more than one large-scale eddy can form inside the canyon; these eddies are vertically aligned and counter-rotate with the adjacent eddies (Liu et al., 2004). This results in even poorer ground-level air quality (Zhong et al., 2015), as the decreasing strength of each eddy with decreasing height makes it increasingly more difficult for any pollutants to be transported upward and removed from the canyon (Li et al., 2008b, Li
et al., 2009).
2.2 Large-eddy simulation 2.2.1 Foundations
Large-eddy simulation is just one of a number of models used to simulate atmospheric flow. Like most of these models, LES numerically solves the dynamic equations of atmospheric motion (which are essentially universal) on a discretised grid. However, the major differences between LES and the other types of model include the spatio-temporal scales of motion typically simulated, and the processes parameterised. For example, general circulation (or global climate) models (GCMs) employ grids that cover the entire globe and are used to simulate the planetary and synoptic scales (i.e. the largest wavelengths of the atmosphere’s kinetic energy spectrum). To resolve these scales, a GCM might typically employ a horizontal grid resolution and model time-step of around 100 km and 30 mins, respectively, whilst the important unresolved processes (subgrid motions, radiation, cloud cover, convection, etc.) must be parameterised. Limited area models (LAMs) employ finer spatio-temporal resolutions to simulate ‘intermediate’ atmospheric motions (i.e. the mesoscales of the kinetic energy spectrum) over a specific region of the globe. Depending on the flow phenomena of interest, the horizontal grid resolution might typically fall within the range 1-50 km and the model time-step between 1-10 min, and the important subgrid processes must again be
17 parameterised. LES, on the other hand, is typically used to simulate even smaller regions of the atmosphere (within the boundary layer), in which the turbulence scales dominate. To resolve boundary-layer-scale eddies, for example, a (3-D) grid resolution and model time-step of around of 100 m and 1 second, respectively, might be required. The simulation domain is usually small enough to elicit the assumption of horizontally homogeneity for many atmospheric processes, thus reducing the number of parameterisations required by the model. For example, simulation of the dry neutral ABL requires no parameterisation for thermal or moisture effects.
LES numerically solves the filtered Navier-Stokes and continuity equations on a discretised model grid. The technique was first proposed by Smagorinsky (1963) in the context of general circulation experiments, and later developed by Deardorff (1970a) for application to the neutral ABL. The LES filter separates the larger eddies, which are resolved by the model, from the smaller eddies, which are not resolved and must therefore be parameterised. Formally, the filtering procedure can be regarded as a convolution of the velocity field,
𝒖(𝒙, 𝑡), with the (normalised) filter kernel, 𝐺, to produce a filtered velocity field, 𝒖̃(𝒙, 𝑡):
𝒖 ̃(𝒙, 𝑡) = 𝐺 ∗ 𝒖(𝒙, 𝑡) = ∫ ∫ 𝒖(𝒙′, 𝑡′)𝐺(𝒙 − 𝒙′, 𝑡 − 𝑡′)𝑑𝑡′𝑑𝒙′ ∞ −∞ ∞ −∞ , (14)
where the symbol ∗ denotes the convolution of one function with another. Application of the LES filter to the governing equations of motion (Eqs. (1) and (2)) gives:
𝜕𝑢̃𝑖 𝜕𝑥𝑖 = 0, (15) 𝜕𝑢̃𝑖 𝜕𝑡 + 𝑢̃𝑗 𝜕𝑢̃𝑖 𝜕𝑥𝑗 = −1 𝜌 𝜕𝑝̃ 𝜕𝑥𝑖 −𝜕𝜏𝑖𝑗 𝜕𝑥𝑗 , (16)
18 where 𝑢̃𝑖 (= 𝑢̃, 𝑣̃, 𝑤̃ for 𝑖 = 1, 2, 3) is now the filtered velocity component in the direction 𝑥𝑖 (= 𝑥, 𝑦, 𝑧 for 𝑖 = 1, 2, 3), 𝑝̃ is the filtered pressure, and 𝜏𝑖𝑗 is the turbulent SGS stress tensor. Again, molecular viscosity is assumed to be negligible compared with the SGS stresses for large-Re flow. Note that the application of the filter to the advection term in Eq. (2) has generated two terms in Eq. (16): the filtered advection term (i.e. the advection of momentum by the resolved velocity field), which is known, and the term involving 𝜏𝑖𝑗, which represents the effects of the unresolved velocity field on the resolved field. Since knowledge of the unresolved velocity field is irrecoverably lost during the filtering process, these effects are unknown and must therefore be parameterised. This is the job of the SGS model, as discussed in the next section (§2.2.2). For the sake of brevity, the tilde (~) is dropped throughout the rest of the thesis (unless stated explicitly); Eqs. (15) and (16) thus become:
𝜕𝑢𝑖 𝜕𝑥𝑖 = 0, (15a) 𝜕𝑢𝑖 𝜕𝑡 + 𝑢𝑗 𝜕𝑢𝑖 𝜕𝑥𝑗 = −1 𝜌 𝜕𝑝 𝜕𝑥𝑖 −𝜕𝜏𝑖𝑗 𝜕𝑥𝑗 . (16a) In addition, the filtered transport equation for a passive scalar 𝐶 can be solved to represent the dispersion of a passive scalar:
𝜕𝐶 𝜕𝑡 + 𝑢𝑗 𝜕𝐶 𝜕𝑥𝑗 = − 𝜕𝜎𝑖 𝜕𝑥𝑖+ 𝑆𝑟𝑐, (17)
where 𝐶 is now the filtered scalar field, and 𝜎𝑗 are the SGS scalar fluxes, which again must be handled by the SGS model.
The scale of the LES filter that separates the larger eddies from the smaller ones should typically lie within the inertial subrange of the TKE spectrum (Geurts, 2004). This range falls between the larger production scales and the smallest dissipation scales (see Figure 2.3,
19 adapted from Garratt (1994)). Eddies within the production scales are typically highly anisotropic and are formed from the mean flow; they thus contain most of the turbulent energy. This energy is then transferred down to smaller scales via the turbulent cascade, until (at the dissipation scales) individual eddies are small enough that their kinetic energy is converted into internal energy by molecular viscosity (Wallace and Hobbs, 2006). Within the inertial subrange, energy is thus neither input into or removed from the flow, but simply passed across the different scales via eddy interactions. These eddies are typically fairly isotropic. Using dimensional analysis, Kolmogorov (1941) was able to show that the inertial subrange of the TKE spectrum follows a behaviour proportional to 𝑘−5/3. This, and other aspects of Kolmogorov theory, later helped steer LES modellers towards successful modelling of the unresolved scales (i.e. those smaller than the LES filter – see §2.2.2, ‘Subgrid-scale modelling’).
Figure 2.3 – Schematic of TKE spectrum
Adapted from Garratt (1994), Fig. 2.1: “Schematic representation of the energy spectrum of turbulence.” The red line indicates a typical ‘cut-off’ wavenumber for LES, which should ideally lie within the inertial subrange. This wavenumber is the inverse of the LES cut-off wavelength ∆, i.e. the length-scale of the LES filter.
20 Due to the chaotic nature of the atmosphere, the time evolution of a given flow-field can only be accurately predicted out to a time-scale that scales on that of the largest individual structures within the flow (e.g. cyclones, boundary-layer eddies, etc.). Thus, unlike numerical weather prediction (NWP) which can be expected to predict, with reasonable accuracy, the evolution of synoptic-scale motions over a few days or so, the time evolution of the atmospheric flows that are typically simulated by LES would only be predictable out to a few seconds or minutes. However, this is typically not the purpose of LES; instead, the technique is used to derive time-averaged statistics (over periods much longer than those of the largest eddies) for a given turbulent flow-field. In fact, whilst the accuracy of an NWP forecast is directly dependent on the accuracy of the initial condition, the statistical data derived from LES are independent of the initial condition. LES models are thus typically validated against time-averaged statistical or theoretical profiles rather than the measured instantaneous data.
2.2.2 Subgrid-scale modelling
It is recalled that the purpose of the SGS model in LES is to parameterise the effects of the unresolved scales on the resolved ones. The majority of SGS models are purely dissipative, implying that they seek to represent the net energy drain (from resolved to unresolved scales) rather than forward and backscatter separately (Leslie and Quarini, 1979). This is typically achieved through a ‘net’ eddy-viscosity, 𝑣sgs, in an analogous way to molecular diffusion. With the popular Smagorinsky (1963) model, the magnitude of 𝑣sgs follows from a local energy balance equation in which quantities on the smallest resolved scales are used to facilitate closure:
𝜏𝑖𝑗 − 1
21 𝑆𝑖𝑗 = 1 2( 𝜕𝑢𝑖 𝜕𝑥𝑗 + 𝜕𝑢𝑗 𝜕𝑥𝑖), (19) 𝑣sgs= (𝐶𝑆Δ)2√2𝑆𝑖𝑗𝑆𝑖𝑗, (20) where 𝛿𝑖𝑗 is the Kronecker delta (𝛿𝑖𝑗 = 1 for 𝑖 = 𝑗, 𝛿𝑖𝑗 = 0 for 𝑖 ≠ 𝑗), 𝐶𝑆 is the so-called Smagorinsky constant, and Δ = (Δ𝑥Δ𝑦Δ𝑧)1/3 is the local grid-scale. Physically, 𝑆𝑖𝑗 (the ‘rate of strain’ tensor) describes the local rate of change of fluid deformation, and is calculated from the resolved velocity field. Thus, in order to close the model, only one parameter (𝐶𝑆) must be prescribed. Typical values of 𝐶𝑆 employed for LES of turbulent atmospheric flows fall within the range 0.1 – 0.2; larger values lead to greater dissipation of energy from the resolved-scale flow, and thus suppress more of the turbulent motions. The isotropic part of the SGS stresses (the 2nd term on the left-hand side of Eq. (18)) is absorbed into the pressure gradient term in Eq. (16). Similarly, the SGS scalar fluxes are modelled using an eddy-diffusivity, 𝛼sgs:
𝜎𝑖 = −𝛼sgs𝜕𝐶
𝜕𝑥𝑖, (21)
𝛼sgs= 𝑣sgs
𝑆𝑐 , (22)
where 𝑆𝑐 is the Schmidt number; another model parameter. Larger values of 𝛼sgs (i.e. smaller 𝑆𝑐) correspond to greater scalar mixing by subgrid motions, thus leading to a more rapidly diffusing scalar within the flow-field. Despite known deficiencies, the Smagorinsky model is often adequate in many simple flows, and remains the most popular choice for SGS modelling due, in part, to its computationally low cost.
The Smagorinsky model is an example of a first-order closure, as it parameterises the (unknown) SGS fluxes directly, thus retaining only three prognostic equations (for 𝑢, 𝑣, 𝑤).
22 Alternatively, it is possible to write down six further prognostic equations for the six unknowns (i.e. the unique components of the tensor 𝑢̃𝑖𝑢𝑗, using the tilde again temporarily). However, these equations contain an even larger number of unknowns, 𝑢̃𝑖𝑢𝑗𝑢𝑘; this pattern continues so that an infinite set of prognostic equations would be required to fully describe a turbulent flow – the so-called closure problem (Keller and Friedmann, 1924). Despite this problem, the sophistication of the SGS model still increases with the number of prognostic equations explicitly solved. Thus, second-order SGS models, which parameterise the third moments (𝑢̃𝑖𝑢𝑗𝑢𝑘) whilst explicitly prognosing the second moments (𝑢̃𝑖𝑢𝑗), typically give more accurate LES than with first-order models. Such models, however, are rarely formulated due to their complexity. One-and-a-half-order SGS models are far more common, e.g. Schumann (1975), Moeng (1984). These models still employ an SGS eddy-viscosity, but solve a further prognostic equation for the SGS energy to calculate its point-wise value; they are thus often referred to as TKE-1.5 models.
Another method for calculating the point-wise values of 𝑣sgs, proposed by Germano et al. (1991), is known as the dynamic SGS modelling approach. Dynamic models use a separate explicit ‘test’ filter, with a width that is slightly larger than that of the LES-filter, to obtain the eddy-viscosity model coefficient (e.g. 𝐶𝑆 in Eq. (20)) measured from the smallest resolved scales. The assumption of scale-invariance is then invoked by applying this value of 𝐶𝑆 directly to the SGS model. Since the resolved scales vary in space and time, so do the calculated values of 𝐶𝑆, hence the reason for the name ‘dynamic model’. As the equation set that must be solved for 𝐶𝑆 is over-determined, this later led Lilly (1992) to propose a minimum least-square error method for obtaining 𝐶𝑆. Disadvantages of the dynamic model include the extra computational effort required over the standard Smagorinsky (i.e. constant coefficient) model, as well as the fact that it is possible to obtain negative values of 𝐶𝑆, which
23 can lead to flow instabilities (Kirkil et al., 2012). Volume averaging of the calculated SGS constants is often performed to reduce the risk of such instabilities, as done, e.g., in the Lagrangian-averaged scale-dependent (LASD) model (Bou-Zeid et al., 2005).
Finally, so-called backscatter SGS models, that do attempt to represent the forward and backscatter processes separately, have also been proposed. These are covered in more detail in §2.3 (‘Backscatter modelling’).
2.2.3 Previous LES of the neutral ABL
LES was first used to simulate the neutral ABL by Deardorff (1970a). This seminal work provided a wealth of information on many aspects of the flow, including the characteristic structure of the large-scale eddies, the effect of varying wind direction on TKE, and variations in the profiles of mean wind, wind shear and other dimensionless turbulence statistics. This was soon followed by the first LES of a dispersing tracer in the neutral ABL (Deardorff, 1972), in which the vertical transportation rate of particles released near the ground was calculated via Lagrangian integrations. Moeng (1984) later continued Deardorff’s pioneering work using a new spectral LES code to systematically investigate different turbulent ABL flows. Mason and Thomson (1987) also revisited LES of the neutral ABL, comparing simulations of different grid resolutions in order to assess the dependence of output statistics on the SGS model. It was also shown that the large-scale eddies are elongated in the direction of the mean wind, and that the characteristic eddy size increases with height above the ground.
In well-resolved regions of the ABL, the energy carried by the SGS motions accounts for only a small portion (the exact value of which depends on the grid resolution adopted) of the total available energy, and previous studies have indicated that simulation performance is largely
24 unaffected by the choice of SGS model in such cases (Mason, 1994). In less well-resolved regions, however, the SGS model carries a more appreciable fraction of the available energy, and the potential consequences of a lack of modelled backscatter can be more severe. Poorly resolved regions in LES should therefore be avoided. However, due to limited computational resources, this is not possible close to solid surfaces, where the characteristic length-scale of the largest eddies becomes increasingly smaller (Mason, 1994).
Numerous LES studies of the ABL have shown that SGS models that do not account for important grid-scale backscatter almost invariably lead to over-prediction of near-surface velocity shear (Mason and Thomson, 1992, Andren et al., 1994, Michioka and Chow, 2008, Kirkil et al., 2012, Talbot et al., 2012). In the neutral surface layer, this is seen as a deviation from the expected logarithmic velocity profile (Eq. (11)), or from the expected constant velocity shear profile (Eq. (12)). Brasseur and Wei (2010) refer to this as the overshoot issue, and reference several further examples of studies in which this issue has been observed. The relevant figure from their paper has been reproduced below (Figure 2.4), showing the different profiles of normalised velocity shear, ΦM, which should equal 1 in the surface layer (see Eq. (13)).
Despite being localised to the near-surface region, any associated simulation deficiencies can be fed up into, and subsequently infect, the larger turbulent length-scales away from the surface, leading to further deterioration in simulation accuracy (Chamecki, 2010). In the surface layer itself, these errors will directly affect the transportation and dispersion of fluxes and scalars, to the detriment of LES-driven dispersion models.
25
Figure 2.4 – Examples of the ‘overshoot’
Taken from Brasseur and Wei (2010), Fig. 1: “Examples of the overshoot in mean shear from previous LES studies: (a) Sullivan et al. (1994), (b) Kosović (1997), (c) Porté-Agel et al. (2000), and (d) Chow et al. (2005). … The shaded regions indicate the surface layer.”
Brasseur and Wei (2010) further showed that this overshoot can be reduced with systematic adjustments to the simulation that include increasing the grid resolution (in all 3 dimensions), allowing for more of the turbulence scales to be explicitly resolved, and thus imposing less reliance on the SGS model itself. However, the resolution required to enter the so-called ‘high accuracy zone’, in which the overshoot becomes negligible, is only computationally feasible for LES of ‘engineering-type’ flows, in which the Reynolds number is typically much lower than in atmospheric flows. This has prompted atmospheric modellers to seek viable alternatives. As it will be seen in §2.3, SGS models that attempt to account for important
26 backscatter in the under-resolved surface layer can help significantly alleviate the overshoot problem in simulations of the neutral ABL.
2.2.4 Previous LES of street canyon flow
Street canyon flow and dispersion have been extensively studied by means of controllable simplified experiments, including reduced-scale wind-tunnel (Meroney et al., 1996, Kastner-Klein and Plate, 1999, Pavageau and Schatzmann, 1999, Brown et al., 2000, Simoëns and Wallace, 2008, Salizzoni et al., 2009, Blackman et al., 2015) and water-channel (Baik et al., 2000, Li et al., 2008a, Di Bernardino et al., 2015) testing, as well as numerical CFD modelling (Baik and Kim, 1999, Baik and Kim, 2002, Liu and Barth, 2002, Walton and Cheng, 2002, Cui et al., 2004, Li et al., 2005, Liu et al., 2005, Cai et al., 2008, Cheng and Liu, 2011a, Michioka et al., 2011, Cai, 2012a, Liu and Wong, 2014). CFD models offer a number of advantages over laboratory experiments, including lower set-up and running costs, significantly better spatial coverage, and the ability to test a variety of urban configurations with relative ease. These CFD models must, however, be well validated to instil confidence in their efficacy. They fall into one of two categories: RANS models, which parameterise all turbulence length-scales in search of the mean flow and dispersion patterns; and LES models, which parameterise only the smallest turbulence length-scales (whilst resolving the larger scales) and retrieve the mean spatial patterns by time-averaging the instantaneous model output record (Li et al., 2006). LES is computationally more expensive than RANS but offers greater simulation accuracy.
In particular, LES is well-equipped to model the dynamics of single-recirculation skimming flow, where a largely isolated primary eddy forms within a street canyon of aspect ratio
𝐻/𝑊 ≈ 1 when the mean wind is perpendicular to the street axis (Oke, 1987). Unlike the
27 and Jeong and Andrews (2002), LES is able to capture important unsteadiness in the roof-level turbulence field (Li et al., 2006). The strengths of LES compared with RANS are also demonstrated in many other studies, e.g., Xie and Castro (2006), Dejoan et al. (2010), Tominaga and Stathopoulos (2010), Salim et al. (2011a) and Salim et al. (2011b). Among the first to apply LES to an individual (reduced-scale) street canyon of unity aspect ratio were Liu and Barth (2002); an analysis of subsequent driven scalar transport showed good agreement between predicted mean concentration profiles within the canyon and measured values. Soon after, Cui et al. (2004) conducted LES within a full-scale street canyon of unity aspect ratio. Mean normalised streamwise velocity, vertical velocity and resolved-scale turbulent kinetic energy (RS-TKE) profiles, generated at five locations across the canyon, gave a noteworthy reproduction of the main features observed in the corresponding wind-tunnel data of Brown et al. (2000). More recently, Cheng and Liu (2011a) and Liu and Wong (2014) utilised larger computing resources to consider 3 and 12 adjacent street canyons, respectively, rather than the one canyon of Cui et al. (2004).
A shared deficiency amongst these LES modelling studies of street canyon dynamics, however, is an under-prediction of the PE intensity within the street canyon. Since the background flow is typically prescribed by applying a constant mass flow rate above the canopy level, the total momentum flux within the street canyon comes entirely from the free-stream flow above it; this deficiency thus indicates insufficient entrainment of high-momentum air across the roof-level shear layer. Given that LES is well validated in its representation of turbulence scales that are not too close to either the domain size or the grid resolution (Mason, 1994), it is likely that the LES models are failing to accurately represent either (or both) the large-scale eddies within the free-stream flow that bring momentum into the street canyon via large ‘sweep’ events (Inagaki et al., 2012), or the small (grid-scale and
28 below) eddies within the roof-level shear layer that mix momentum down into the street canyon via turbulent diffusion (Letzel et al., 2008). The LES domain size limits the size of the large-scale eddies in the free-stream flow; their vertical extent is restricted by the domain lid height and their horizontal extent confined to half the domain width (assuming periodic lateral boundary conditions). Interestingly, however, Cheng and Liu (2011a) and Liu and Wong (2014) observed no significant change in their normalised velocity profiles (and, by inference, in their simulated PE intensity) compared with Cui et al. (2004), despite modelling more than one adjacent street canyon (the domain size was also increased in the span-wise and vertical directions). This will have allowed for significantly larger free-stream eddies to form, which suggests that a significant portion of their simulation degradation was attributable to the misrepresentation of the grid-scale (and smaller) eddies.
Improving simulation accuracy of the smaller turbulence scales at roof-level is a challenging task. In this region, the narrow shear layer that exists due to the sharp reduction in streamwise velocity between the fast-moving free-stream air above the street canyon and the relatively slow-moving air within it, continually produces small (yet energetic) eddies through Kelvin-Helmholtz instability (Louka et al., 2000). Very fine grid spacing is therefore required in order to explicitly resolve much of this roof-level turbulence. The LES simulations performed by Letzel et al. (2008) suggest that a resolution of at least 100 across-canyon grid points is required in order to explicitly resolve these Kelvin-Helmholtz waves. However, their associated large computational demands necessitated a rather low domain lid height of only 1.5𝐻. Indeed, the computational resources available to most industrial end-users are typically far smaller than those available to research institutions, and resolution sacrifices are often unavoidable.
